Question

How do you find a power series representation for `f(x)= 9/(1-3x)^2` and what is the radius of convergence?

Answer 1

See below

Explanation:

`3/(1-3x)^2 = d/(dx)(1/(1-3x))` but for `abs (3x) < 1`

`1/(1-3x) = lim_(n->oo)sum_(k=0)^n3^kx^k` hence

`f(x) = 9/(1-3x)^2 = 3sum_(k=0)^oo 3^kx^k`

convergent for `absx < 1/3`